Tamim made the most of his milestone 50th test, scoring 71 in the first innings and top-scoring with 78 in the second.
Bangladesh skipper Mushfiqur Rahim suggested after the match that the Australians had attempted to run down the clock at the end of the first session on day four.
"They had aggression, but we also showed that Bangladesh can be an aggressive side," Mushfiqur said. "They saw it in our body language, apart from how we did with the bat and ball.
"If you noticed towards the end of the first session, [Australian batsman Glenn] Maxwell tried to ensure that there was no more overs by taking up five of the six minutes.
"The Australian team were on the back foot, trying not to play an over. I thought it is a huge message."
Emboldened by their remarkable triumph in Dhaka, despite being ranked ninth in the world, Mushfiqur said his side had proven themselves capable of standing up to the Australians.
"They found out that we are no longer the Bangladesh of old, that we now have the players who can now change the course of a match from any position," he said. "But we should be careful of not becoming too complacent.
"It is hard to get the Australian team under pressure, so we must take the opportunity to win the series."